MinT - Best Known (54, 54+89, s)-Nets in Base 9

Best Known (54, 54+89, s)-Nets in Base 9

(54, 54+89, 81)-Net over F₉ — Constructive and digital

Digital (54, 143, 81)-net over F₉, using

t-expansion [i] based on digital (32, 143, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(54, 54+89, 182)-Net over F₉ — Digital

Digital (54, 143, 182)-net over F₉, using

t-expansion [i] based on digital (50, 143, 182)-net over F₉, using
- net from sequence [i] based on digital (50, 181)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
    - a curve from the manYPoints database [i]

(54, 54+89, 2563)-Net in Base 9 — Upper bound on s

There is no (54, 143, 2564)-net in base 9, because

1 times m-reduction [i] would yield (54, 142, 2564)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 3186 548971 795594 879358 389855 981592 638841 520457 436848 587756 763500 246820 764086 830171 956952 175478 647718 092028 624789 333449 641280 825830 027905 > 9¹⁴² [i]